Solve 5x^2-24x-7

$$5 x ^ { 2 } - 24 x - 7$$

Factor

$$5\left(x-\frac{12-\sqrt{179}}{5}\right)\left(x-\frac{\sqrt{179}+12}{5}\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$5x^{2}-24x-7=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 5\left(-7\right)}}{2\times 5}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 5\left(-7\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-24\right)±\sqrt{576-20\left(-7\right)}}{2\times 5}$$

Multiply -20 times -7.

$$x=\frac{-\left(-24\right)±\sqrt{576+140}}{2\times 5}$$

Add 576 to 140.

$$x=\frac{-\left(-24\right)±\sqrt{716}}{2\times 5}$$

Take the square root of 716.

$$x=\frac{-\left(-24\right)±2\sqrt{179}}{2\times 5}$$

The opposite of -24 is 24.

$$x=\frac{24±2\sqrt{179}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{24±2\sqrt{179}}{10}$$

Now solve the equation x=\frac{24±2\sqrt{179}}{10} when ± is plus. Add 24 to 2\sqrt{179}.

$$x=\frac{2\sqrt{179}+24}{10}$$

Divide 24+2\sqrt{179} by 10.

$$x=\frac{\sqrt{179}+12}{5}$$

Now solve the equation x=\frac{24±2\sqrt{179}}{10} when ± is minus. Subtract 2\sqrt{179} from 24.

$$x=\frac{24-2\sqrt{179}}{10}$$

Divide 24-2\sqrt{179} by 10.

$$x=\frac{12-\sqrt{179}}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{12+\sqrt{179}}{5} for x_{1} and \frac{12-\sqrt{179}}{5} for x_{2}.

$$5x^{2}-24x-7=5\left(x-\frac{\sqrt{179}+12}{5}\right)\left(x-\frac{12-\sqrt{179}}{5}\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 5

$$x ^ 2 -\frac{24}{5}x -\frac{7}{5} = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = \frac{24}{5} rs = -\frac{7}{5}$$

Two numbers r and s sum up to \frac{24}{5} exactly when the average of the two numbers is \frac{1}{2}*\frac{24}{5} = \frac{12}{5}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = \frac{12}{5} - u s = \frac{12}{5} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{7}{5}

$$(\frac{12}{5} - u) (\frac{12}{5} + u) = -\frac{7}{5}$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{144}{25} - u^2 = -\frac{7}{5}$$

Simplify the expression by subtracting \frac{144}{25} on both sides

$$-u^2 = -\frac{7}{5}-\frac{144}{25} = -\frac{179}{25}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{179}{25} u = \pm\sqrt{\frac{179}{25}} = \pm \frac{\sqrt{179}}{5}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =\frac{12}{5} - \frac{\sqrt{179}}{5} = -0.276 s = \frac{12}{5} + \frac{\sqrt{179}}{5} = 5.076$$

Evaluate

$$5x^{2}-24x-7$$